We go to the third method in numerical integration, the Simpson’s Rule. Here, instead of actually using line segments in getting an approximate of a curve, we now use parabolas. We first divide an interval \left [a, b\right ] into n subintervals of equal length. The figure below shows the curve that will be approximated.

Here, we represent the length h= \Delta x= \frac{b-a}{n}, but this time we’ll assume that n is an even number. For each consecutive pair of intervals we approximate the curve y=f(x) \geq 0 by a parabola as shown in the above figure.

Now if we let y_i=f(x_i ), then P_i (x_i,y_i ) is the point on the curve lying above x_i. A typical parabola passes through three consecutive points, P_i, P_{i+1}, and P_{i+2}. For us to be able to solve the problem easier, let us consider the case where x_0= -h, x_1=0 ,and x_2=h. We can illustrate this as follows,

From our elementary algebra, we know that the a parabola through the points P_0, P_1, and P_2 take the form of y=ax^2+ bx+c. We can therefore calculate the area under the parabola by integrating the equation from x = -h to x = h which is

Although we have derived this approximation for the case in which x \geq 0 , it is a reasonable approximation for any continuous function f and is called Simpson’s Rule after the English mathematician Thomas Simpson (1710–1761). Note the pattern of coefficients: 1, 4, 2, 4, 2, 4, 2, . . . , 4, 2, 4, 1.